Actions and identities on set partitions

Details Actions and identities on set partitions Actions and identities on set partitions

2011-07-21

arxiv.org/abs/1107.4173v3

Mathematics

Combinatorics

28 pages; v3: material revised with additional final section

Scientific paper

A labeled set partition is a partition of a set of integers whose arcs are labeled by nonzero elements of an abelian group $A$. Inspired by the action of the linear characters of the unitriangular group on its supercharacters, we define a group action of $A^n$ on the set of $A$-labeled partitions of an $(n+1)$-set. By investigating the orbit decomposition of various families of set partitions under this action, we derive new combinatorial proofs of Coker's identity for the Narayana polynomial and its type B analogue, and establish a number of other related identities. In return, we also prove some enumerative results concerning Andr\'e and Neto's supercharacter theories of type B and D.

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